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Theorem crngohomfo 28953
Description: The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
Hypotheses
Ref Expression
crnghomfo.1  |-  G  =  ( 1st `  R
)
crnghomfo.2  |-  X  =  ran  G
crnghomfo.3  |-  J  =  ( 1st `  S
)
crnghomfo.4  |-  Y  =  ran  J
Assertion
Ref Expression
crngohomfo  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e. CRingOps )

Proof of Theorem crngohomfo
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 754 . 2  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e.  RingOps )
2 foelrn 5970 . . . . . . . 8  |-  ( ( F : X -onto-> Y  /\  y  e.  Y
)  ->  E. w  e.  X  y  =  ( F `  w ) )
32ex 434 . . . . . . 7  |-  ( F : X -onto-> Y  -> 
( y  e.  Y  ->  E. w  e.  X  y  =  ( F `  w ) ) )
4 foelrn 5970 . . . . . . . 8  |-  ( ( F : X -onto-> Y  /\  z  e.  Y
)  ->  E. x  e.  X  z  =  ( F `  x ) )
54ex 434 . . . . . . 7  |-  ( F : X -onto-> Y  -> 
( z  e.  Y  ->  E. x  e.  X  z  =  ( F `  x ) ) )
63, 5anim12d 563 . . . . . 6  |-  ( F : X -onto-> Y  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  ( E. w  e.  X  y  =  ( F `  w )  /\  E. x  e.  X  z  =  ( F `  x ) ) ) )
7 reeanv 2992 . . . . . 6  |-  ( E. w  e.  X  E. x  e.  X  (
y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  <->  ( E. w  e.  X  y  =  ( F `  w )  /\  E. x  e.  X  z  =  ( F `  x ) ) )
86, 7syl6ibr 227 . . . . 5  |-  ( F : X -onto-> Y  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  E. w  e.  X  E. x  e.  X  ( y  =  ( F `  w )  /\  z  =  ( F `  x ) ) ) )
98ad2antll 728 . . . 4  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  E. w  e.  X  E. x  e.  X  ( y  =  ( F `  w )  /\  z  =  ( F `  x ) ) ) )
10 crnghomfo.1 . . . . . . . . . . . . . 14  |-  G  =  ( 1st `  R
)
11 eqid 2454 . . . . . . . . . . . . . 14  |-  ( 2nd `  R )  =  ( 2nd `  R )
12 crnghomfo.2 . . . . . . . . . . . . . 14  |-  X  =  ran  G
1310, 11, 12crngocom 28948 . . . . . . . . . . . . 13  |-  ( ( R  e. CRingOps  /\  w  e.  X  /\  x  e.  X )  ->  (
w ( 2nd `  R
) x )  =  ( x ( 2nd `  R ) w ) )
14133expb 1189 . . . . . . . . . . . 12  |-  ( ( R  e. CRingOps  /\  (
w  e.  X  /\  x  e.  X )
)  ->  ( w
( 2nd `  R
) x )  =  ( x ( 2nd `  R ) w ) )
15143ad2antl1 1150 . . . . . . . . . . 11  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( w ( 2nd `  R ) x )  =  ( x ( 2nd `  R ) w ) )
1615fveq2d 5802 . . . . . . . . . 10  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
w ( 2nd `  R
) x ) )  =  ( F `  ( x ( 2nd `  R ) w ) ) )
17 crngorngo 28947 . . . . . . . . . . 11  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
18 eqid 2454 . . . . . . . . . . . 12  |-  ( 2nd `  S )  =  ( 2nd `  S )
1910, 12, 11, 18rngohommul 28923 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
w ( 2nd `  R
) x ) )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
2017, 19syl3anl1 1267 . . . . . . . . . 10  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
w ( 2nd `  R
) x ) )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
2110, 12, 11, 18rngohommul 28923 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  X  /\  w  e.  X ) )  -> 
( F `  (
x ( 2nd `  R
) w ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2221ancom2s 800 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
x ( 2nd `  R
) w ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2317, 22syl3anl1 1267 . . . . . . . . . 10  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
x ( 2nd `  R
) w ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2416, 20, 233eqtr3d 2503 . . . . . . . . 9  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( ( F `  w ) ( 2nd `  S ) ( F `
 x ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
25 oveq12 6208 . . . . . . . . . 10  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( y ( 2nd `  S ) z )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
26 oveq12 6208 . . . . . . . . . . 11  |-  ( ( z  =  ( F `
 x )  /\  y  =  ( F `  w ) )  -> 
( z ( 2nd `  S ) y )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2726ancoms 453 . . . . . . . . . 10  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( z ( 2nd `  S ) y )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2825, 27eqeq12d 2476 . . . . . . . . 9  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( ( y ( 2nd `  S ) z )  =  ( z ( 2nd `  S
) y )  <->  ( ( F `  w )
( 2nd `  S
) ( F `  x ) )  =  ( ( F `  x ) ( 2nd `  S ) ( F `
 w ) ) ) )
2924, 28syl5ibrcom 222 . . . . . . . 8  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( ( y  =  ( F `  w
)  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
3029ex 434 . . . . . . 7  |-  ( ( R  e. CRingOps  /\  S  e.  RingOps 
/\  F  e.  ( R  RngHom  S ) )  ->  ( ( w  e.  X  /\  x  e.  X )  ->  (
( y  =  ( F `  w )  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) ) )
31303expa 1188 . . . . . 6  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
w  e.  X  /\  x  e.  X )  ->  ( ( y  =  ( F `  w
)  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) ) )
3231adantrr 716 . . . . 5  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( ( w  e.  X  /\  x  e.  X )  ->  (
( y  =  ( F `  w )  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) ) )
3332rexlimdvv 2951 . . . 4  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( E. w  e.  X  E. x  e.  X  ( y  =  ( F `  w
)  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
349, 33syld 44 . . 3  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  (
y ( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
3534ralrimivv 2911 . 2  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  A. y  e.  Y  A. z  e.  Y  ( y ( 2nd `  S ) z )  =  ( z ( 2nd `  S ) y ) )
36 crnghomfo.3 . . 3  |-  J  =  ( 1st `  S
)
37 crnghomfo.4 . . 3  |-  Y  =  ran  J
3836, 18, 37iscrngo2 28945 . 2  |-  ( S  e. CRingOps 
<->  ( S  e.  RingOps  /\  A. y  e.  Y  A. z  e.  Y  (
y ( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
391, 35, 38sylanbrc 664 1  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e. CRingOps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2798   E.wrex 2799   ran crn 4948   -onto->wfo 5523   ` cfv 5525  (class class class)co 6199   1stc1st 6684   2ndc2nd 6685   RingOpscrngo 24013    RngHom crnghom 28913  CRingOpsccring 28942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fo 5531  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-map 7325  df-rngo 24014  df-com2 24049  df-rngohom 28916  df-crngo 28943
This theorem is referenced by: (None)
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